
Construct a set \(\mathcal {M} \subseteq V\)  satisfying [1]} by using the Moser-Tardos algorithm.
This can be parallelized by the parallel Moser-Tardos algorithm [2]}. The local lemma condition (REF ) is strong enough to be compatible with the parallel Moser-Tardos algorithm in constructing \(\mathcal {M} \subseteq V\)  satisfying [1]}.

Simulate the Glauber dynamics for the distribution \(\mu _\mathcal {M}\)  projected from \(\mu \)  onto \(\mathcal {M}\) .
This is a single-site dynamics on the complete graph on \(\mathcal {M}\) , with local update distributions \(\lbrace P_v^\tau \rbrace \)  specified as \(P_v^\tau =\mu _v^{\tau _{\mathcal {M} \setminus \lbrace v\rbrace }}\) , where \(\tau \)  is the current assignment on \(\mathcal {M}\) .
By [1]}, all assignments on \(\mathcal {M}\)  having positive measure in \(\mu _\mathcal {M}\) , \(\lbrace P_v^\tau \rbrace \)  are well-defined.
By [1]}, this chain mixes in \(t_{\mathsf {mix}}^{\mathsf {D}}(\epsilon )=O(n\log \frac{n}{\epsilon })\)  steps (which translates to \(t_{\mathsf {mix}}^{\mathsf {C}}(\epsilon )=O(\log \frac{n}{\epsilon })\)  continuous time by (REF )), since it satisfies a path coupling condition [1]},
which also implies cond:main with \(C=1\)  in \(\ell _\infty \)  norm.
Applying thm:parallel with \(T=t_{\mathsf {mix}}^{\mathsf {C}}(1/\mathrm {poly}(n))=O(\log n)\) , alg:efficient-simulation simulates this chain within \(O(\log n)\)  depth and \(\tilde{O}(|\mathcal {M}|^2)=\tilde{O}(n^2)\)  processors, assuming oracles for evaluating \(\mu _v^{\tau _{\mathcal {M} \setminus \lbrace v\rbrace }}\) .
Drawing from \(\mu _v^{\tau _{\mathcal {M} \setminus \lbrace v\rbrace }}\)  is done by [1]}.
At each update of the chain, with high probability, the clauses satisfied already by the current assignment on \({\mathcal {M} \setminus \lbrace v\rbrace }\)  disconnect the CNF into components of small sizes.
A rejection sampling can be applied to the component containing \(v\) , with each trial succeeding with probability \(\ge n^{-2^{-20}}\) .

At last, after the Glauber dynamics on \(\mu _\mathcal {M}\)  mixes, the random assignment on \(\mathcal {M}\)  is extended to all variables.
This is also done by [1]} that draws from \(\mu _{V\setminus \mathcal {M}}^{\tau _{\mathcal {M}}}\) .

